172.52/44.00 YES 172.52/44.01 172.52/44.01 Problem: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 172.52/44.01 Proof: 172.52/44.01 DP Processor: 172.52/44.01 DPs: 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 b#(x1) -> c#(x1) 172.52/44.01 b#(x1) -> a#(c(x1)) 172.52/44.01 b#(x1) -> c#(a(c(x1))) 172.52/44.01 TRS: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 TDG Processor: 172.52/44.01 DPs: 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 b#(x1) -> c#(x1) 172.52/44.01 b#(x1) -> a#(c(x1)) 172.52/44.01 b#(x1) -> c#(a(c(x1))) 172.52/44.01 TRS: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 graph: 172.52/44.01 b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> b#(a(x1)) 172.52/44.01 b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) -> b#(x1) -> c#(a(c(x1))) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) -> b#(x1) -> a#(c(x1)) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) -> b#(x1) -> c#(x1) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> c#(a(c(x1))) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> a#(c(x1)) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) -> b#(x1) -> c#(x1) 172.52/44.01 a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) 172.52/44.01 SCC Processor: 172.52/44.01 #sccs: 1 172.52/44.01 #rules: 4 172.52/44.01 #arcs: 12/36 172.52/44.01 DPs: 172.52/44.01 b#(x1) -> a#(c(x1)) 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 TRS: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 Arctic Interpretation Processor: 172.52/44.01 dimension: 2 172.52/44.01 usable rules: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 interpretation: 172.52/44.01 [b#](x0) = [1 0]x0 + [3], 172.52/44.01 172.52/44.01 [a#](x0) = [-& 0 ]x0 + [0], 172.52/44.01 172.52/44.01 [-& 0 ] [1 ] 172.52/44.01 [c](x0) = [0 -&]x0 + [-&], 172.52/44.01 172.52/44.01 [0 -&] [2] 172.52/44.01 [b](x0) = [1 0 ]x0 + [3], 172.52/44.01 172.52/44.01 [0] 172.52/44.01 [a](x0) = x0 + [0] 172.52/44.01 orientation: 172.52/44.01 b#(x1) = [1 0]x1 + [3] >= [0 -&]x1 + [0] = a#(c(x1)) 172.52/44.01 172.52/44.01 a#(b(x1)) = [1 0]x1 + [3] >= [-& 0 ]x1 + [0] = a#(x1) 172.52/44.01 172.52/44.01 a#(b(x1)) = [1 0]x1 + [3] >= [1 0]x1 + [3] = b#(a(x1)) 172.52/44.01 172.52/44.01 a#(b(x1)) = [1 0]x1 + [3] >= [1 0]x1 + [3] = b#(b(a(x1))) 172.52/44.01 172.52/44.01 [0] 172.52/44.01 a(x1) = x1 + [0] >= x1 = x1 172.52/44.01 172.52/44.01 [0 -&] [2] [0 -&] [2] 172.52/44.01 a(b(x1)) = [1 0 ]x1 + [3] >= [1 0 ]x1 + [3] = b(b(a(x1))) 172.52/44.01 172.52/44.01 [0 -&] [2] [1] 172.52/44.01 b(x1) = [1 0 ]x1 + [3] >= x1 + [1] = c(a(c(x1))) 172.52/44.01 172.52/44.01 [1] 172.52/44.01 c(c(x1)) = x1 + [1] >= x1 = x1 172.52/44.01 problem: 172.52/44.01 DPs: 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 TRS: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 Restore Modifier: 172.52/44.01 DPs: 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 TRS: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 EDG Processor: 172.52/44.01 DPs: 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 TRS: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 graph: 172.52/44.01 a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) 172.52/44.01 a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1)) 172.52/44.01 a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(b(a(x1))) 172.52/44.01 SCC Processor: 172.52/44.01 #sccs: 1 172.52/44.01 #rules: 1 172.52/44.01 #arcs: 3/9 172.52/44.01 DPs: 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 TRS: 172.52/44.01 a(x1) -> x1 172.52/44.01 a(b(x1)) -> b(b(a(x1))) 172.52/44.01 b(x1) -> c(a(c(x1))) 172.52/44.01 c(c(x1)) -> x1 172.52/44.01 Usable Rule Processor: 172.52/44.01 DPs: 172.52/44.01 a#(b(x1)) -> a#(x1) 172.52/44.01 TRS: 172.52/44.01 172.52/44.01 Arctic Interpretation Processor: 172.52/44.01 dimension: 4 172.52/44.01 usable rules: 172.52/44.01 172.52/44.01 interpretation: 172.52/44.01 [a#](x0) = [0 -& -& -&]x0, 172.52/44.01 172.52/44.01 [1 0 0 0] [0] 172.52/44.01 [0 0 0 0] [0] 172.52/44.01 [b](x0) = [1 0 0 0]x0 + [0] 172.52/44.01 [0 0 0 0] [0] 172.52/44.01 orientation: 172.52/44.01 a#(b(x1)) = [1 0 0 0]x1 + [0] >= [0 -& -& -&]x1 = a#(x1) 172.52/44.01 problem: 172.52/44.01 DPs: 172.52/44.01 172.52/44.01 TRS: 172.52/44.01 172.52/44.01 Qed 172.52/44.01 EOF